Good dynamics versus bad kinematics: Is entanglement needed for quantum computation?

We study the role of entanglement in quantum computation. We consider the case of a pure state contaminated by "white noise." This framework arises, for example, in pseudopure state implementations of quantum computing using NMR. We analyze quantum computational protocols which aim to solve exponential classical problems with polynomial resources and ask whether or not entanglement of the pseudopure states is needed to achieve this aim. We show that, for a large class of such protocols, including Shor's factorization, entanglement is necessary. We also show that achieving entanglement is not sufficient: If the state is sufficiently noisy, exponential resources are needed even if entanglement is present.